First off, I'm assuming that you're using American in the broader sense of the western hemisphere. That being said there were a few important differences between slavery in the Americas and slavery before it. First, slavery in America came to be based purely on race. The relatively small number of slaves in Europe that had existed before were taken primarily through war, and were almost always of a different religion, rather than necessarily a different race. In fact, there's pretty good evidence that the idea of "race" hadn't even been codified yet. Second, there's simply the question of scale. Slavery in the Americas skyrocketed slavery from a relatively insignificant portion of the laboring population, primarily on sugar plantations in the Mediterranean, into the dominant form of labor in the area. Third, before American slavery, most forms of slavery were not chattel slavery. By chattel slavery, I mean a state of being in which the enslaved were considered purely the property of the owner, who could do pretty much whatever he wished with it. Slaves had few, or no, rights, little official power in the community, would always be slaves (unless they were freed by their master), and any children they had would also be slaves. This is very different from almost any other form of slavery preceding it. In the Ottoman Empire, for instance, slaves could achieve quite high rank in the government and wield quite a bit of power. Also, in the Comanche Empire, the children of slaves could be accepted into the tribe with little or no bias against them.

Now, each European empire in the New World had their own cultural differences that led to different varieties of slavery, but they tended to share these common themes, unofficially in some cases. I can't really speak for modern slavery, since I'm not an expert on it.

Sources: Philip Curtin, Rise and Fall of the Plantation Complex,